Character Tables
tables
This chapter describes operations for character tables of finite groups.

Operations for characters (or, more general, class functions) are described in Chapter .

For a description of the &GAP; Library of Character Tables, see the separate manual for the &GAP; package `CTblLib`.

Several examples in this chapter require the &GAP; Character Table Library to be available. If it is not yet loaded then we load it now.

gap> LoadPackage( "ctbllib" ); true
Some Remarks about Character Theory in &GAP;
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History of Character Theory Stuff in GAP

&GAP; provides functions for dealing with group characters since the version &GAP; 3.1, which was released in March 1992. The reason for adding this branch of mathematics to the topics of &GAP; was (apart from the usefulness of character theoretic computations in general) the insight that &GAP; provides an ideal environment for developing the algorithms needed. In particular, it had been decided at Lehrstuhl D für Mathematik that the `CAS` system (a standalone Fortran program together with a database of character tables, see ) should not be developed further and the functionality of `CAS` should be made available in &GAP;. The background was that extending `CAS` (by new Fortran code) had turned out to be much less flexible than writing analogous &GAP; library code.

For integrating the existing character theory algorithms, &GAP;'s memory management and long integer arithmetic were useful as well as the list handling –it is an important feature of character theoretic methods that questions about groups are translated into manipulations of lists; on the other hand, the datatype of cyclotomics (see Chapter ) was added to the &GAP; kernel because of the character theory algorithms. For developing further code, also other areas of &GAP;'s library became interesting, such as permutation groups, finite fields, and polynomials.

The development of character theory code for &GAP; has been supported by several DFG grants, in particular the project Representation Theory of Finite Groups and Finite Dimensional Algebras (until 1991), and the Schwerpunkt Algorithmische Zahlentheorie und Algebra (from 1991 until 1997). Besides that, several Diploma theses at Lehrstuhl D were concerned with the development and/or implementation of algorithms dealing with characters in &GAP;.

The major contributions can be listed as follows. The arithmetic for the cyclotomics data type, following , was first implemented by Marco van Meegen; an alternative approach was studied in the diploma thesis of Michael Scherner (see ) but was not efficient enough; later Martin Schönert replaced the implementation by a better one. The basic routines for characters and character tables were written by Thomas Breuer and Götz Pfeiffer. The lattice related functions, such as , , and , were implemented by Ansgar Kaup (see ). Functions for computing possible class fusions, possible power maps, and table automorphisms were written by Thomas Breuer (see ). Functions for computing possible permutation characters were written by Thomas Breuer (see ) and Götz Pfeiffer (see ). Functions for computing character tables from groups were written by Alexander Hulpke (Dixon-Schneider algorithm, see ) and Hans Ulrich Besche (Baum algorithm and Conlon algorithm, see ). Functions for dealing with Clifford matrices were written by Ute Schiffer (see ). Functions for monomiality questions were written by Thomas Breuer and Erzsébet Horváth.

Since then, the code has been maintained and extended further by Alexander Hulpke (code related to his implementation of the Dixon-Schneider algorithm) and Thomas Breuer.

Currently &GAP; does not provide special functionality for computing Brauer character tables, but there is an interface to the `MOC` system (see ), and the &GAP; Character Table Library contains many known Brauer character tables.

Creating Character Tables
<#Include Label="[10]{ctbl}"> <#Include Label="CharacterTable"> <#Include Label="BrauerTable"> <#Include Label="CharacterTableRegular"> <#Include Label="SupportedCharacterTableInfo"> <#Include Label="ConvertToCharacterTable">

Character Table Categories
<#Include Label="IsNearlyCharacterTable"> <#Include Label="InfoCharacterTable"> <#Include Label="NearlyCharacterTablesFamily">

Conventions for Character Tables

The following few conventions should be noted. The class of the identity element is expected to be the first one; thus the degree of a character is the character value at position 1. The trivial character of a character table need not be the first in the list of irreducibles. Most functions that take a character table as an argument and work with characters expect these characters as an argument, too. For some functions, the list of irreducible characters serves as the default, i.e, the value of the attribute ; in these cases, the value is automatically computed if it was not yet known. For a stored class fusion, the image table is denoted by its value; each library table has a unique identifier by which it can be accessed (see in the manual for the &GAP; Character Table Library), tables constructed from groups get an identifier that is unique in the current &GAP; session.

The Interface between Character Tables and Groups
<#Include Label="[2]{ctbl}"> <#Include Label="UnderlyingGroup:ctbl"> <#Include Label="ConjugacyClasses:ctbl"> <#Include Label="IdentificationOfConjugacyClasses"> <#Include Label="CharacterTableWithStoredGroup"> <#Include Label="CompatibleConjugacyClasses">

Operators for Character Tables
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Attributes and Properties for Groups and Character Tables
<#Include Label="[4]{ctbl}"> <#Include Label="CharacterDegrees"> <#Include Label="Irr"> <#Include Label="LinearCharacters"> <#Include Label="OrdinaryCharacterTable"> <#Include Label="[5]{ctbl}">

Attributes and Properties only for Character Tables
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Normal Subgroups Represented by Lists of Class Positions
<#Include Label="[8]{ctbl}"> <#Include Label="ClassPositionsOfNormalSubgroups"> <#Include Label="ClassPositionsOfAgemo"> <#Include Label="ClassPositionsOfCentre:ctbl"> <#Include Label="ClassPositionsOfDirectProductDecompositions"> <#Include Label="ClassPositionsOfDerivedSubgroup"> <#Include Label="ClassPositionsOfElementaryAbelianSeries"> <#Include Label="ClassPositionsOfFittingSubgroup"> <#Include Label="ClassPositionsOfLowerCentralSeries"> <#Include Label="ClassPositionsOfUpperCentralSeries"> <#Include Label="ClassPositionsOfSupersolvableResiduum"> <#Include Label="ClassPositionsOfPCore"> <#Include Label="ClassPositionsOfNormalClosure">

Operations Concerning Blocks
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Other Operations for Character Tables
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<#Include Label="Index!for_character_tables"> <#Include Label="IsInternallyConsistent!for_character_tables"> <#Include Label="IsPSolvableCharacterTable"> <#Include Label="IsClassFusionOfNormalSubgroup"> <#Include Label="Indicator"> <#Include Label="NrPolyhedralSubgroups"> <#Include Label="ClassMultiplicationCoefficient:ctbl"> <#Include Label="ClassStructureCharTable"> <#Include Label="MatClassMultCoeffsCharTable">

Printing Character Tables
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Computing the Irreducible Characters of a Group

Several algorithms are available for computing the irreducible characters of a finite group G. The default method for arbitrary finite groups is to use the Dixon-Schneider algorithm (see ). For supersolvable groups, Conlon's algorithm can be used (see ). For abelian-by-supersolvable groups, the Baum-Clausen algorithm for computing the irreducible representations (see ) can be used to compute the irreducible characters (see ).

These functions are installed in methods for , but explicitly calling one of them will not set the value of G.

<#Include Label="IrrDixonSchneider"> <#Include Label="IrrConlon"> <#Include Label="IrrBaumClausen"> <#Include Label="IrreducibleRepresentations"> <#Include Label="IrreducibleRepresentationsDixon">

Representations Given by Modules

This section describes functions that return certain modules of a given group. (Extensions by modules can be formed by the command .)

<#Include Label="IrreducibleModules"> <#Include Label="AbsoluteIrreducibleModules"> <#Include Label="RegularModule">

The Dixon-Schneider Algorithm
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Advanced Methods for Dixon-Schneider Calculations
irreducible characters

The computation of irreducible characters of very large groups may take quite some time. On the other hand, for the expert only a few irreducible characters may be needed, since the other ones can be computed using character theoretic methods such as tensoring, induction, and restriction. Thus &GAP; provides also step-by-step routines for doing the calculations. These routines allow one to compute some characters and to stop before all are calculated. Note that there is no safety net: The routines (being somehow internal) do no error checking, and assume the information given is correct.

When the info level of if positive, information about the progress of splitting is printed. (The default value is zero.)

<#Include Label="DixonRecord"> <#Include Label="DixonInit"> <#Include Label="DixontinI"> <#Include Label="DixonSplit"> <#Include Label="BestSplittingMatrix"> <#Include Label="DxIncludeIrreducibles"> <#Include Label="SplitCharacters"> <#Include Label="IsDxLargeGroup">

Components of a Dixon Record

The Dixon record D returned by stores all the information that is used by the Dixon-Schneider routines while computing the irreducible characters of a group. Some entries, however, may be useful to know about when using the algorithm interactively, see .

group
the group G of which the character table is to be computed,
conjugacyClasses
classes of G (all characters stored in the Dixon record correspond to this arrangement of classes),
irreducibles
the already known irreducible characters (given as lists of their values on the conjugacy classes),
characterTable
the value of G (whose irreducible characters are not yet known),
ClassElement( D, el )
a function that returns the number of the class of G that contains the element el.
An Example of Advanced Dixon-Schneider Calculations

First, we set the appropriate info level higher.

gap> SetInfoLevel( InfoCharacterTable, 1 );

for printout of some internal results. We now define our group, which is isomorphic to PSL_4(3).

gap> g:= PrimitiveGroup(40,5); PSL(4, 3) gap> Size(g); 6065280 gap> d:= DixonInit( g );; #I 29 classes #I choosing prime 65521 gap> c:= d.characterTable;;

After the initialisation, one structure matrix is evaluated, yielding smaller spaces and several irreducible characters.

gap> DixonSplit( d ); #I Matrix 9,Representative of Order 3,Centralizer: 5832 #I Dimensions: [ 1, 2, 1, 4, 12, 1, 1, 2, 1, 2, 1 ] #I Two-dim space split #I Two-dim space split #I Two-dim space split 9

In this case spaces of the listed dimensions are a result of the splitting process. The three two dimensional spaces are split successfully by combinatoric means.

We obtain several irreducible characters by tensor products and notify them to the Dixon record.

gap> asp:= AntiSymmetricParts( c, d.irreducibles, 2 );; gap> ro:= ReducedCharacters( c, d.irreducibles, asp );; gap> Length( ro.irreducibles ); 3 gap> DxIncludeIrreducibles( d, ro.irreducibles );

The tensor products of the nonlinear characters among each other are reduced with the irreducible characters. The result is split according to the spaces found, which yields characters of smaller norms, but no new irreducibles.

gap> nlc:= Filtered( d.irreducibles, i -> i[1] > 1 );; gap> t:= Tensored( nlc, nlc );; gap> ro:= ReducedCharacters( c, d.irreducibles, t );; ro.irreducibles; [ ] gap> List( ro.remainders, i -> ScalarProduct( c, i, i) ); [ 2, 2, 4, 4, 4, 4, 13, 13, 18, 18, 19, 21, 21, 36, 36, 29, 34, 34, 42, 34, 48, 54, 62, 68, 68, 78, 84, 84, 88, 90, 159, 169, 169, 172, 172, 266, 271, 271, 268, 274, 274, 280, 328, 373, 373, 456, 532, 576, 679, 683, 683, 754, 768, 768, 890, 912, 962, 1453, 1453, 1601, 1601, 1728, 1739, 1739, 1802, 2058, 2379, 2414, 2543, 2744, 2744, 2920, 3078, 3078, 4275, 4275, 4494, 4760, 5112, 5115, 5115, 5414, 6080, 6318, 7100, 7369, 7369, 7798, 8644, 10392, 12373, 12922, 14122, 14122, 18948, 21886, 24641, 24641, 25056, 38942, 44950, 78778 ] gap> t:= SplitCharacters( d, ro.remainders );; gap> List( t, i -> ScalarProduct( c, i, i ) ); [ 2, 2, 4, 2, 2, 4, 4, 3, 6, 5, 5, 9, 9, 4, 12, 13, 18, 18, 20, 18, 20, 24, 26, 32, 32, 16, 42, 59, 69, 69, 72, 72, 36, 72, 78, 78, 84, 122, 117, 127, 117, 127, 64, 132, 100, 144, 196, 256, 456, 532, 576, 679, 683, 683, 754, 768, 768, 890, 912, 962, 1453, 1453, 1601, 1601, 1728, 1739, 1739, 1802, 2058, 2379, 2414, 2543, 2744, 2744, 2920, 3078, 3078, 4275, 4275, 4494, 4760, 5112, 5115, 5115, 5414, 6080, 6318, 7100, 7369, 7369, 7798, 8644, 10392, 12373, 12922, 14122, 14122, 18948, 21886, 24641, 24641, 25056, 38942, 44950, 78778 ]

Finally we calculate the characters induced from all cyclic subgroups and obtain the missing irreducibles by applying the LLL-algorithm to them.

gap> ic:= InducedCyclic( c, "all" );; gap> ro:= ReducedCharacters( c, d.irreducibles, ic );; gap> Length( ro.irreducibles ); 0 gap> l:= LLL( c, ro.remainders );; gap> Length( l.irreducibles ); 13

The LLL returns class function objects (see Chapter ), and the Dixon record works with character values lists. So we convert them to a list of values before feeding them in the machinery of the Dixon-algorithm.

gap> l.irreducibles[1]; Character( CharacterTable( PSL(4, 3) ), [ 640, E(13)^7+E(13)^8+E(13)^11, E(13)^4+E(13)^10+E(13)^12, E(13)^2+E(13)^5+E(13)^6, E(13)+E(13)^3+E(13)^9, 0, -8, 0, -8, 0, 0, 0, 0, 0, -8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1 ] ) gap> l:=List(l.irreducibles,ValuesOfClassFunction);; gap> DxIncludeIrreducibles( d, l ); gap> Length( d.irreducibles ); 29 gap> Length( d.classes ); 29

It turns out we have found all irreducible characters. As the last step, we obtain the irreducible characters and tell them to the group. This makes them available also to the character table.

gap> irrs:= DixontinI( d );; #I Total:1 matrices,[ 9 ] gap> SetIrr(g,irrs); gap> Length(Irr(c)); 29 gap> SetInfoLevel( InfoCharacterTable, 0 );
Constructing Character Tables from Others
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Sorted Character Tables
<#Include Label="CharacterTableWithSortedCharacters"> <#Include Label="SortedCharacters"> <#Include Label="CharacterTableWithSortedClasses"> <#Include Label="SortedCharacterTable"> <#Include Label="ClassPermutation">

Automorphisms and Equivalence of Character Tables
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Storing Normal Subgroup Information
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